Thunderbird
Close matchFreeFor iCloud Mail on Linux, add the account to Thunderbird and read it in a normal mail app instead of a browser tab.
Your files: Mail stays on Apple's servers and downloads again into Thunderbird.
Does iCloud run on Linux?
Verdict: Partly
Partly. There is no iCloud app for Linux, but icloud.com works in a browser and covers mail, photos, files, contacts, calendar and notes. The automatic sync folder is what you lose.
Apple makes iCloud for Windows and for its own devices, and nothing for Linux. There is no sign of one coming.
What you keep is icloud.com. Sign in with your Apple Account in Firefox or Chrome and you get Mail, Photos, iCloud Drive, Contacts, Calendar, Notes, Reminders, Find Devices, Hide My Email and the Pages, Numbers and Keynote apps. For reading mail, checking the calendar and getting at a file or a photo, that is enough.
What you lose is the quiet background part. On Windows, iCloud for Windows puts an iCloud Drive folder and a Photos folder in File Explorer and keeps them up to date on their own, and it fills in passwords in your browser. None of that happens on Linux. Files and photos become something you upload and download by hand in a browser tab, which is a real step down if you were used to dropping a file in a folder and having it appear on your iPhone.
Your account and everything in it is untouched. Your iPhone and iPad carry on as normal. If the sync folder matters to you, the usual answer is to keep using iCloud in the browser for Apple things and set up a separate service for the folder you actually work in.
What to use instead of iCloud
Nextcloud
Different way of workingFree with paid extrasGives you back the sync folder that iCloud for Windows provided, with files, calendar and contacts. It is a different account and service, not a way into iCloud.
Your files: Your files must be copied across from iCloud once. After that they sync both ways on Linux.
Syncthing
Different way of workingFreeKeeps a folder matched between your own computers with no cloud account at all. No phone photo backup and no web access.
Your files: Copy your files in once and they stay matched between your devices.
Dropbox
Different way of workingFree with paid extrasHas a proper Linux sync folder if what you want is that behaviour back, on a service that supports Linux officially.
Your files: Files must be moved out of iCloud into Dropbox by hand the first time.
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